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arXiv:1307.3878 · Journal of Mathematical Physics

Demkov-Kunike models with decay

M. B. Kenmoe, A. B. Tchapda, Lukong Cornelius Fai

Published Dec 1, 2016• 7 citations

Abstract

Exact analytical solutions to the dissipative time-dependent Schrödinger equation are obtained for a decaying two-state system with decay rates Γ1 and Γ2 for the levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only t0 = 0 or t0 = − ∞ as for non-decaying models and may be useful for the experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses, while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.

DOI: 10.1063/1.4972289

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Exact analytical solutions to the dissipative time-dependent Schrödinger equation are obtained for a decaying two-state system with decay rates Γ1 and Γ2 for the levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only t0 = 0 or t0 = − ∞ as for non-decaying models and may be useful for the experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses, while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.

The paper presents an analytic study of a two-state system which interacts with an external pulsed field. The excited state is supposed to decay out of the system with a decay rate \ensuremath{\Gamma}. We present two exact analytic solutions, viz., the finite Landau-Zener model and the Demkov-Kunike model. We also derive an approximate formula for arbitrary pulse shapes and detunings in the limit of a large decay rate. It gives the exact result for the original Landau-Zener model and the correct asymptotics for large \ensuremath{\Gamma} in the general case. We conclude that the independence of the final ground-state population on the decay rate, recently established for the original Landau-Zener model [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)], is only valid for the latter and not for any other model. The results show that the population loss ultimately decreases at large decay rates which can be viewed as a quantum analog of the overdamped oscillator.

The Demkov-Kunike (DK) model, characterized by a time-dependent Rabi coupling

J\phantom{\rule{0.16em}{0ex}}\text{sech}(t/T)

and on-site detuning ${\mathrm{\ensuremath{\Delta}}}_{0}+{\mathrm{\ensuremath{\Delta}}}_{1}tanh(t/T)$, has one of the most general forms of an exactly solvable two-state quantum system and therefore it provides a paradigm for coherent manipulations of a qubit's quantum state. Despite its extensive application in noise-free cases, the exploration of the noisy DK model remains limited. Here we extend the coherent DK model to take into account a noisy coupling term $J\ensuremath{\rightarrow}{J}_{\text{noisy}}(t)$. We consider colored Markovian noise sources represented by telegraph noise and Gaussian noise. We present exact solutions for the survival probability

{Q}_{\text{DK}}^{\text{noisy}}

of the noisy DK model, namely, the probability of the system to remain in its initial state. For slow telegraph noise, we identify parameter regimes where the survival probability

{Q}_{\text{DK}}^{\text{noisy}}

is suppressed rather than enhanced by noise. In contrast, for slow Gaussian noise, the noise always enhances the survival probability

{Q}_{\text{DK}}^{\text{noisy}}

, due to the absorption of noise quanta across the energy gap. This study not only complements the existing research on the noisy Landau-Zener model but also provides valuable insights into the control of two-level quantum systems.

We present an analytically exactly solvable two-state quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The exact solution is expressed in terms of associated Legendre functions. An interesting feature of this model is that the excitation probability does not vanish, except for zero pulse area or zero detuning; this feature is attributed to the asymmetric pulse shape. Two limiting cases are considered. When the coupling rises very slowly, it is nearly linear and the tanh model reduces to the shark model introduced earlier. When the coupling rises very quickly, the tanh model reduces to the Rabi model, which assumes a rectangular pulse shape and hence a sudden switch on. Because of its practical significance, we have elaborated the asymptotics of the solution in the Rabi limit, and we have derived the next terms in the asymptotic expansion, which deliver the corrections to the amplitude and the phase of the Rabi oscillations due to the finite rise time of the coupling.

Abstract We present a novel time-dependent two-state model that describes a constant-amplitude level-crossing field configuration, where the frequency detuning varies within a finite interval. A distinctive feature of this configuration is that the resonance crossing always occurs asymmetrically in time, making it an asymmetric version of the second Demkov-Kunike model. The general solution of the problem is expressed in terms of two independent irreducible linear combinations of the Gauss hypergeometric functions. We analyze the asymptotes of the solution in terms of corresponding quasi-energies and calculate the final transition probability in the case when the system starts from the first quasi-energy state.

We present an analytically exactly soluble two-state model, in which a hyperbolic-secant-shaped pulsed interaction has a phase jump of $\ensuremath{\phi}$ at the time of its maximum. The detuning has a constant part and a hyperbolic-tangent chirp term. For $\ensuremath{\phi}=0$, this model reduces to the Demkov-Kunike model, which in turn contains as particular cases three other well-known models: the Rosen-Zener, Allen-Eberly, and Bambini-Berman models. A nonzero $\ensuremath{\phi}$ induces dramatic changes in the transition probability, ranging from complete population inversion to complete population return. The analytic results are particularly instructive in the adiabatic limit and demonstrate that complete population inversion can always occur for a suitable choice of $\ensuremath{\phi}$. The jump phase $\ensuremath{\phi}$ can therefore be used as a control parameter for the two-state transition probability.

Abstract We introduce a new non-symmetric-in-time version of the second Demkov–Kunike model. This is a constant-amplitude level-crossing field configuration, where the detuning varies in a restricted frequency interval and crosses the resonance in a non-symmetric way with respect to the crossing time point. We provide the solution of the two-state problem for this field configuration using irreducible linear combinations of hypergeometric functions and calculate the transition probability at time going to infinity for the scenario where the system begins in the first quasi-energy state.